Could someone intuitively explain to me Ohm's law?
I understand what voltage is and how it is the electric potential energy and that it is the integral of the electric field strength etc. I also understand that current is the rate at which charge flows at a specific point in the circuit, and I get that resistivity is the opposite of conductivity and that it's analogous to friction in some ways, but I cannot at all get the whole picture and connect the 3 together.
In addition to the other answers, here is something for the intuition:
$$V=RI$$
More "pressure" $V$ (more correctly: higher "pressure" difference from one side to the other) is required to keep the flow $I$ of charges constant when the flow is resisted by $R$. A thin wire has higher resistance than a thick wire, $R=\rho L/A$, analogous to a "bottleneck" in a traffic jam.
Think of plumbing for a close analogy. Voltage is how hard you are pushing, and current is how much flows. The relationship writes itself: why would you get more or less flow from the same pump? The measure of how much effort is used to get flow (it makes more sense as the reciprical: how much flows for a unit of effort) is the interesting property, and it's defined to fit in that relation exactly.
Conductivity is how much current flows for a unit voltage. It's defined by that observation. The blockage, friction, constriction, or whatnot of the pipe makes less flow for the same effort. E.g. a narrow pipe, or a curved section of corrigated pipe, will have a higher resistance.
Have you looked at Drude's model? I was taught something like that back at school and have kept it in mind as a intuitive way of understanding it.
We want to understand why the current (rate of flow of charge) should be linear with the potential difference.
The Drude idea is, as you noted, related to friction.
Firstly, the EM field is linear in the potential difference. This generates a force on the electrons in the conductor.
The electrons then accelerate with the field. Were their path unimpeded they would constantly accelerate. Instead, they 'collide' with atoms in the structure of the conductor and 'bounce' off. In steady state, the rate of specific impulse transferred from bounces has to balance the force due to the EM field.
The key observation by Drude is that the velocity of the electron at the time of collision will be directly proportional to the strength of the field.
Please be clear that this is only an intuitive model, albeit one that gives surprising insight despite being fairly crude.
If you go in depth of resistivity, it will be easy to get through the point. Voltage is the reason for the movements (flow) of electrons that produce current (charge divided by time). If you have many electrons and atoms in the way (like barriers, like when you are running in a crowd!) they reduce the rate of charge flow. Now it is clear that if there are more barriers (depends on things like the geometry, shape, material) you will get less current.
If you set up a circuit with any component (not just resistors) connected to a voltage source, you will find out that the current which will flow through the component depends on the voltage. In most cases, the higher the voltage the higher the current you will get.
Conversely, you can ask: how large a voltage does it take to get a certain current through the component? Again, this depends on the current you want to flow through through the component. To get small currents, small voltages suffice. For large currents, you need large voltages.
This qualitative observation is what almost all electrical components have in common. But examining the situation quantitavely will lead to this kind of question: How many volts do I need per ampere to have my desired current flow through the component. The answer depends on the component, and the physical quantity is called resistance.
Example: a two ohms resistor. Two ohms means you need two volts per ampere. So if you want 10 amperes to flow through the resistor you need 20 Volts.
I've always looked at it as an analogy to a "drop in potential energy". ${M*g*H}$ is the potential energy of a mass that is held a distance $H$ above the ground. If it drops halfway, it then has half the potential energy.
If the current passes through a resistor, the voltage drop or "potential energy drop" is equal to $I*R$. You now have less "driving force" to push the desired current through the next resistor if there is one.
In other words, the battery can only push so much current through the existing resistor system.
This illustration uses a pipe filled with water to represent a wire conducting electricity.
Amps, aka current, can be thought of as volume of water and is controlled by the size of the wire (or tube here, represented as ohms aka resistance). Volts would be the water pressure, or intensity of electricity.
So a wire's size limits the amps, just as a pipe's size limits water.
